Surfactant Limited Aggregation of Hydrophobic Molecules in

HAL Id: jpa-00248446https://hal.archives-ouvertes.fr/jpa-00248446

Submitted on 1 Jan 1997

HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.

Surfactant Limited Aggregation of HydrophobicMolecules in Water

Hélène Lannibois, Anwar Hasmy, Robert Botet, Olivier Chariol, BernardCabane

To cite this version:Hélène Lannibois, Anwar Hasmy, Robert Botet, Olivier Chariol, Bernard Cabane. Surfactant LimitedAggregation of Hydrophobic Molecules in Water. Journal de Physique II, EDP Sciences, 1997, 7 (2),pp.319-342. �10.1051/jp2:1997128�. �jpa-00248446�

https://hal.archives-ouvertes.fr/jpa-00248446

https://hal.archives-ouvertes.fr

J. Phys. II FYance 7 (1997) 319-342 FEBRUARY 1997, PAGE 319

Surfactant Limited Aggregation of Hydrophobic Moleculesin Water

H4lAne Lannibois (~), Anwar Hasmy (~), Robert Botet (~,*),Olivier Aguerre Chariol (3) and Bernard Cabane (~)

(~ Equipe mixte CEA RP, Rhbne-Poulenc, 93308 Aubervilliers, France

(~) Laboratoire de Physique des Solides, Universit6 Paris-Sud, Bhtiment 510,

91405 Orsay, France

(~) Rhbne-Poulenc, 93308 Aubervilliers, France

(Received 28 July1996, received jn final form 11 October 1996, accepted 25 October 1996)

PACS.82.70.Kj Emulsions and suspensionsPACS.82.20.Wt Computational modeling; simulation

PACS.05.40.+j Fluctuation phenomena, random processes, and Brownian motion

Abstract. The precipitation in water of hydrophobic molecules has been studied in presence

of added surfactants. Amorphous particles grow through aggregation of clusters of hydrophobicmolecules; the growth is terminated by adsorption of surfactant. The particle qizes vary accord-

ing to the concentrations of hydrophobic molecules and of surfactant molecules. Two regimeshave been found for the use of surfactant Inolecules: at low surfactant concentrations, an efficient

reg1nle where all surfactant nlolecules are adsorbed on the surfaces of the growing particles; at

high surfactant concentrations, a wasteful reg1nle, where excess surfactant molecules are left in

water. Attempts to reduce the particle sizes by adding increasing anlounts of surfactant become

inefficient at sortie point where nlost of the added surfactant renlains in water. These results are

explained by a kinetic aggregation model which s1nlulates the conlpetition between aggregationof hydrophobic nlolecules and adsorption of surfactant. The results of exper1nlents are well re-

produced by s1nlulations where aggregation is allowed to proceed un1nlpeded for a time r, and

then adsorption of the surfactant starts. In these conditions, particle sizes are deternlined by

the rate of aggregation and by the value of this t1nle delay.

1. Introduction

Many drugs currently under development are made of large hydrophobic molecules. In order

to reach the active sites, these molecules must be carried across an aqueous environment.

However, the amount of drug that can be dissolved and transported in the aqueous medium

is much too low for practical purposes: typical solubilities for large hydrophobic molecules

range from 10~8 g/g to 10~5 g/g. The classical way to achieve this transport is to dissolve

the hydrophobic molecules into carrier particles, I.e. supramolecular objects which may be

dispersed in water. Typical carriers are polymeric particles, liposomes, micelles [I-4]. Still,the amount of drug that can be dissolved in most carriers is small, about 10$l of the carriers

mass. Therefore there is a need to design a carrier free delivery system, where particles of

(*) Author for correspondence (e-nlail: botettllps.u-psud.fr)

© Les iditions de Phvsioue 1997

320 JOURNAL DE PHYSIQUE II N°2

pure drug could be covered with a small amount of surfactant and dispersed in water. The

requirements are that the particle size should be smaller than 0.2 ~J, the specific surface area

should be of the orier of100 m2 /g, and the amount of surfactant should be as low as possihle.Very fine dispersions may be obtained, either by fragmentation [5-8], or by growth processes

[9,10]. In fragmentation, particles of hydrophobic products may be produced by emulsification

of the product in water, if it is a liquid, or grinding it down to a very small size, if it is a solid.

In growth processes, the hydrophobic product is dissolved in a solvent that is miscible with

water, and then the solution is mixed with water, causing the insoluble molecules to precipitate.

It is then necessary to control the precipitation process to obtain colloidal particles instead of

a macroscopic solid.

In all these processes, surfactants are needed, either to help the fragmentation (emulsificationof grinding)

or to prevent aggregation of the fragmented particles, or to control growth. The

problem is that the necessary amounts of surfactant arelarge, sometimes comparable with the

amount of hydrophobic product. This reduces the advantage of such delivery systems over the

traditional carriers, and may cause a non specific toxicity of the formula. In this paper, we

address the problem of the use of surfactants for one particular case: dispersions made through

a precipitation process.

We considera

"model" hydrophobic product made of molecules that are nearly insoluble

in water (10~8g/g). This hydrophobic product is dissolved in a solvent that is miscible with

water, and the solution is mixed with a large excess of water. The hydrophobic molecules are

instantly desolvated. Because they are insoluble in water, they stick to each other whenever

they meet, and dissociation reactions do not occur. In such conditions, there is no nucleation

barrier, and aggregates of hydrophobic molecules grow immediately through a Diffusion Limited

Cluster-cluster Aggregation (DLCA) mechanism [11-13].If this aggregation is allowed to continue, the aggregates will become so large that they will

reach macroscopic sizes and fall out of solution (precipitation). However, surfactants may be

added in order to control this growth. Indeed, surfactant molecules may adsorb on the growingaggregates through their hydrophobic tails. Hydrophobic surfaces that are fully covered with

surfactant may not stick to each other; hence, aggregates that are saturated with surfactant

molecules will no longer recombine with other aggregate8. In this way, a stable dispersion of

nanometric aggregates may be obtained.

Th18 description of a po180ned aggregation proces8 suggests that the amount of surfactant

needed to control the final sizes of the aggregates should be the amount which covers their

surfaces. As will be shown below, the aggregates are often spherical, amorphous particles, be-

cause their interior is platicized by residual solvent and water. Thereforea monolayer coverage

by adsorbed surfactant would require a ratio of surfactant volume (Vs) to hydrophobic volume

(VH) volume given by:~ ~~~

R ~~~

wherer is the thickness of the surfactant layer and R the radius of the particles. Typically,

the desired particle diameters are 100 nm, and thus the ratio Vs/l~H should be of the order of

0.06. However, a general finding is that the amount of surfactant needed to control the growthof aggregates in such conditions is vastly in excess of a monolayer coverage. This inefficient use

of surfactants may have two possible origins: either the surfactant molecules are not available

near the particle surfaces during the aggregation process, or they are available, but they do

not protect the surfaces efficiently.In the following, we report the formation of nanoparticles by aggregation of hydrophobic

molecules in presence of surfactants; different conditions have been set for the competition

between aggregation and poisoning by surfactants, and the resulting particle sizes have been

N°2 SURFACTANT LIMITED AGGREGATION 321

measured. In order to interpret these data, we have designed a kinetic model where hydrophobicmolecules and surfactants (or poisons)

are moved at random on a lattice. The process of

poisoned aggregation has been reproduced with a set of rules that determine the stickingprobabilities of all species. This simulation shows two regimes for the use of surfactants:

an "efficient" regime, where the surfactant molecules are used efficiently, but the aggregationproceeds to rather large sizes, and a "wasteful" regime, where a vast excess of surfactant is used

to stop the aggregation at an earlier stage. The comparison with experimental results shows

that the preparation of nanoparticles is usually conducted in the "wasteful" regime where

a large fraction of the surfactant remains in the aqueous phase. Dispersions of nanometric

particles (sizes below 100 nm) cannot be obtained in the "efficient" regime, unless the surfactant

adsorbs more efficiently on the growing aggregates than the hydrophobic molecules themselves.

2. Materials

The hydrophobic product used in most precipitation experiments was cholesteryl acetate. This

product was chosen because it has a very low solubility in water (2 x10~8 g/g),

a 8ubstantial

solubility in polar solvents that are m18cible with water (e.g. 0.034 g/g in acetone at 23 °C),and becau8e it resembles a number of drugs currently in development. It is a solid imp

=

lls °C) which can be obtained in a number of crystal structures [14-16], or in an amorphou8

or microcrystalline 8tate.

We also used a liquid hydrophobic product, hexadecane imp=

18 °C), in order to comparethe sizes of particles prepared through precipitation and through emulsification. The solubilityof hexadecane in water is I x

10~l~ g lg.For neutron scattering experiments, intended to determine the location of the surfactant, we

used deuterated polystyrene to make the cores of the hydrophobic nanoparticles. This polymer

was a generous gift from Claude Picot at the Institut Charles Sadron; it had an average molar

mass AI=

15000 g/mol. It is noted hereafter PSD.

The surfactants used in this study were an ethoxylated fatty alcohol with 12 carbons in the

linear alkyl chain and 5 oxyethylene groups, hereafter named C12E5, and a diblock copolymermade of a styrene block (molar

mass 1000) and an oxyethylene block (molar mass 1000),hereafter named PS-POE. This polymer was obtained from Goldschmidt (type VPSE 1010).

When these surfactants are adsorbed from a micellar solution on a hydrophobic surface, theyform monolayers where the area per surfactant molecule is 0.48 nm~ for C12E5 and 1.28 nm~

for PS-POE; the thickness of these layers are 0.5 nm and 2.5 nm, respectively.The hydrophobic molecules were initially dissolved in polar solvents which were acetone (for

cholesteryl acetate with PSPOE), ethanol (for hexadecane and C12E5) and tetrahydrofuran(THF) (for PSD and PS-POE).

The variables which define the composition of the mixtures are cH, initial concentration of

hydrophobic molecules in the solvent, and cs, initial concentration of surfactant in the solvent.

Both are expressed in g/g. It was found that the outcome of the precipitation is mainly

determined by the value of the ratio cs/cH. Typical values for the composition were: cH,10~3

to 3 x10~~ g/g; ratio cs/CH, 5 x

10~~ to 10; final concentration of hydrophobic molecules in

water, 10~5 to 3 x10~~ g/g; final concentration of solvent in water, 10~~ g/g.

3. Methods

3. I. PRECIPITATION. The precipitation of hydrophobic molecules in water was performed as

follows. First, the hydrophobic product and the surfactant were dissolved in the polar solvent.

Then the solution was injected into a large volume of water. At the moment of injection,


the local supersaturation of hydrophobic molecules was on the order of 105. This very highsupersaturation caused immediate aggregation of the hydrophobic molecules, as described in

the introduction. The aggregation stopped at a stage that was determined by the relative

concentrations of hydrophobic molecules, cH, and of surfactant, cs. The final dispersion had

colloidal stability in all ca8es, excepted for hexadecane droplets covered with a very small

amount of surfactant.

3.2. QuAsi ELASTIC LIGHT SCATTERING. The sizes of the particles in the final dispersion

were calculated from their motions measured through Quasi Elastic Light Scattering (QELS)[17,18]. It was found that the sizes were in the range 50 to 500 nm, for which this method is

adequate.

3. 3. SMALL ANGLE NEUTRON SCATTERING. The 8tructures of the particles were examined

through SANS [17,18]. Neutrons are scattered by differences in scattering length between

nuclei in the particles and nuclei in the continuous phase. Good contrast between particles and

continuous phase was achieved using D2U as the continuous phase. This contrast was used to

determine whether the particles were porous or dense. Alternatively, the hydrophobic molecules

and water were deuterated, while the surfactant remained hydrogenated. This contrast was

used to determine the location and the structure of the surfactant within the particles. Twc-

dimensional scattering patterns were obtained on the instrument Dll at ILL. These patterns

were subsequently radially averaged to yield scattering curves of intensity, I, vs. magnitude,Q, of the scattering vector. Q is related to the wavelength, I, of incident neutrons and to the

scattering angle, 9, by:

Q=

ISin

Ill12)

The general features of scattering curves for independent particles dispersed in a homogeneouscontinuous phase are as follows [19]:(a) at Q -+ 0 the intensity, Io, is proportional to particle mass concentration and to the

square of the difference in density of scattering length, Ap, between D2U and the protonatedmolecules in the particles. Hence, the average content of the particles may be determined;

(b) at low values, the rays scattered by nuclei at opposite ends of the particle interfere

destructively. In this regime, the curvature of the scattering curve measures the average radius

of gyration, Rg, of the particles. This quantity may be determined by fitting the measured

intensity curve to an expansion formula. For globular particles the best expansion is the Guinier

formula:

~2j~2IIQ)"Ioexp ~~ j3)

(c) at higher Q values, the intensity decay reflects interferences at shorter distances within each

particle, I-e- their internal structure. For dense globular particles (no internal structure), the

intensity follows Porod's law, from whidh the surface area A of the particles may be determined:

1(Q)G3

AQ~~ (4)

For flat particles or hollow shells, the destructive interference is not as strong, and the decay

is only according to Q~~i for rod-like particles it is Q~~ For bushy or porous particles with

fractal dimension di the decay follows the power law Q~df Thus, from the slope of the intensitydecay in a log-log plot, the dimensionality of the particles may be determined.


Q~

q.

d1,

'

,~llQ '~

~~l/Lm $~f~

it~i ""

~

Fig. 1. TEM 1nlage of a cholesteryl acetate dispersion in water. Uranyl acetate was added to

the dispersion, then it was dried in air on a collodion nlenlbrane before observation; this caused the

aggregation of nlost particles, and the recrystallization of a few.

3.4. ELECTRON MicRoscoPY. The particle shapes and sizes were also observed throughCryo-Transmission Electron Microscopy (CRYO-TEM) [20]. Thin liquid films were prepared

on perforated carbon films supported on 3 mm electron microscope grids. A specimen was

prepared by applying a small (about 3 ~Jl) drop onto the grid, blotting most of it to the desired

thickness (under 200 nm) and plunging it into liquid ethane at its melting point. This ultra-

fast cooling caused vitrification of the liquid phase, I.e., specimens became solid-like (vitreous)without change of phase that leads to structural rearrangement. These cryo-specimens were

stored under liquid nitrogen, and transferred to the cooling-holder (Gatan 626) of the TEM

(JEOL 2000FX), where they were equilibrated at -170 ° C, and examined with an acceleration

voltage of 100 kV.

3.5. ComPosiTioN VARIABLES. All these measurements were performed in order to studyhow surfactants limit the aggregation process and the loss of surface area that results from

the aggregation. The results are expressed in two ways: first, as an average particle volume

iv; second, as a normalized surface area, I.e. the total surface area of the dispersion divided

by the surface area that would be created if all the surfactant molecules were used to cover

the particle surfaces. These quantities are measured in the final dispersions, made at different

values of the composition variables cH, cs, orcs/cHi the laws for these variations are then

compared with the laws predicted by the kinetic growth models.

4. Experimental Results

4.I. OBSERVATION oF THE DisPERsioNs THROUGH ELECTRON MicRoscoPY. In a first

step, cholesteryl acetate dispersions were dried on a collodion membrane aid examined throughTEM. The image (Fig. I) shows a collection of globular particles, with sizes in the 100 nm

range, and a broad distribution of sizes. It can also be seen that extensive aggregation and


fit,"l'l

I potJ

'~G~

[~ 0.2 /Lm

Fig. 2. CRYO-TEM 1nlage of a cholesteryl acetate dispersion in water with PS-POE as asurfactant

(cH"

10~~ g/g and cs/cH=

0.1). The fi1nl of vitrified water appears as a unifornl grey background.This fi1nl is held across a perforation in a carbon fi1nl, which is also seen. The cholesteryl acetate

particles are in the vitrified water film, pushed by the meniscus against the edge of the fi1nl. The

1nlportant features are the good dispersion of the particles in water, their globular shapes and the

absence of any crystallinity.

some recrystallization have occurred as a result of drying. All subsequent examination of

the dispersions were made through CRYO-TEM. The image shown in Figure 2 shows particlesconfined in the meniscus of the vitrified water film, near the edge of one of the perforations in the

carbon film. The particles appear non-aggregated, with an amorphous structure and globularshapes. This was confirmed in every observation through CRYO-TEM. A measurement of

the particle sizes (diameters)was performed on an image where the particles were uniformly

dispersed in the vitrified water film; the resulting size distribution is shown in Figure 3.

4.2. SYSTEMATIC STUDY oF PARTICLE SizEs. In a first set of experiments, the hydrophobicmolecules were cholesteryl acetate and the surfactant was the diblock copolymer PS-POE. Both

were dissolved in acetone, and the solution was precipitated in water. The initial concentrations

were: cholesteryl acetate, 10~~ g/g; PS-POE, 5 x10~~ to 10~~ g/g; the final concentrations

were 100 times smaller. A colloidal dispersion was obtained in every case. The sizes of the

particles in the dispersion were measured immediately through QELS, and then again at later

times. It was found that these sizes remained the same over more than onemonth. Figure 4

shows the average particle volumes calculated from these radii, at cs/cH"

I, for different

values of the initial concentration cH of cholesteryl acetate in acetone.

The radii vary between 15 and 30 nm, and the volumes between 14 x 10~ and l10 x10~ nm3,

for the initial concentration cu =

10~~ g/g. They are always larger than the volumes that


o.06

0.04

~w

c

0.02

~~~0 lo 20 30 40

s (nm)

Fig. 3. Distribution of particle sizes (diameters) ns nleasured on a CRYO-TEM 1nlage (cH=

10~~ g/g and cs/cH"

1). The 1nlage contained 1000 particles, unifornlly distributed across the fi1nl

of vitrified water. The particles were selected individually and their diameters s were then nleasured

automatically by a conlputer progranl.

would be obtained if all the surfactant had been used to limit aggregation by covering the

particle surfaces: at cs/cH=

I, the particle volume corresponding to an efficient use of the

surfactant would be 4000 nm3. At high dilution, the particle volumes approach the limit of

efficient use of surfactant, and they rise away from it when the concentration cH of hydrophobicmolecules is increased. The law for this rise was found to be the same at all compositions cs/cH

of this system; it is:

v~v C~ CH. (5)

Other experiments were performed with hexadecane as a hydrophobic molecule, and C12E5

as a surfactant, both dissolved in ethanol. At high surfactant concentration (cs/cH"

I and

cs/CH"

0.4), the measured volumes were larger than those of cholesteryl acetate particles, and

much larger than the volumes calculated for an efficient use of surfactant; again, they follow

the law described in equation (5). However, at low surfactant concentration (cs/cH"

0.04),two successive regimes were observed: at large dilutions (cH < 10~~ g/g) a regime of efficient

use of the surfactant (denoted by arrows in Fig. 4), and at higher concentrations, again the

regime of wasteful use of the surfactant (rising straight line in the log-log representation of

Fig. 4).

In summary, the particle volumes are generally larger than those predicted for an efficient

use of the surfactant, and they rise with the concentration of hydrophobic molecules in the

initial solution, according to equation (5).We have also measured systematically the particle volumes obtained with the same initial

dilution cH but different values of the ratio cs/cH (Fig. 5). The data for hexadecane and C12E5show a fast decrease of the average volumes at cs/cH < 0.05, indicating that the surfactant

is used efficiently, followed by a slow decrease at cs/cH > 0.05, indicating that the additional

surfactant is mostly wasted.


io~

tl

~

io~ '

nD

57

E10

>

Io

io~

io~

cH

Fig.

ispersions ade precipitation of solutions ontaining a givenhydrophobic

product and

surfactant; the ratio cs/cH of surfactant tonlolecules

has been kept

overalldilution

of theolution,

measured by cH, has been changed. Hollow cholesteryl

acetate andS-POE,

at cs/cH " 1.Filled squares: hexadecane

and C12E5 at cs/cH

squares: exadecane and 12E5 atcs/cH

" 0.04. The arrowindicates

the alues ofthe volunles

calculated for an efficient use ofthe surfactants,

and the filled lines are a linear fit

The slope of the data is 1 in log-log scales, as predicted by a kinetic nlodel.

In order to put these notions of efficient or wasteful use of the surfactant on a quantitativebasis, it is useful to calculate the amount of particle surface that has been protected from

aggregation by the adsorption of surfactant. The surface area of the particles may be calculated

from their radii, assuming a globular shape. If the surfactant was used efficiently, this surface

area would equal the surface area of a monolayer containing all the surfactant molecules. Thus

the ratio of both quantities is an indicator of the effectiveness of the surfactant. The values of

this ratio are presented in Figure 6.

For hexadecane droplets covered by C12E5, we choose a surface area of 0.47 nm~ per sur-

factant molecule [21,22]. The calculated ratios show a regime of efficient use up to cs/cH=

0.05, followed by a wasteful regime. For cholesteryl acetate particles covered by PS-POE, we

choose a surface area of1.28 nm~ per surfactant molecule [23]. The calculated ratios show a

regime where, surprisingly, more surface is kept than what is covered by the surfactant. In

fact, cholesteryl acetate particles can be obtained in the absence of any added surfactant. This

is because cholesteryl acetate contains ionic impurities that cause the particles to repel each in

any medium where ionic dissociation is possible. Therefore the cholesteryl acetate dispersions

are always in the regime of excess surfactant.

In summary, it is not possible to obtain particles of arbitrarily small volumes by addingsufficient amounts of surfactant: in such cases, only a snlall fraction of the surfactant is efficient

in controlling the aggregation. This result cannot be improved unless one understands the fate

of the wasted surfactant; this is one of the goals of the next section.


~~9

tl

n

~

aio "

~

n©1f

~S ~

~ ~"

C l~~

>~

o

io~ ~

o

O

io~~ ~io io io

C~/C~

Fig. 5. Effect of surfactant concentration cs on the average particle volumes iv. Each set of data

corresponds to dispersions made from the precipitation of solutions containing a given hydrophobicproduct and surfactant; the overall dilution of the solution, measured by cH, has been kept a constant,but the ratio cs/cH of surfactant to hydrophobic nlolecules has been changed. Hollow dianlonds:

Cholesteryl acetate and PS-POE, at cH =0.01 g/g. Filled squares: hexadecane and C12E5 at cH =

0.04 g/g. Hollow squares: hexadecane and C12E5 at cH =0.1 g/g. The lines are a guide to the eye.

4.3. STRUCTURES OF PARTICLES AND LOCATION OF THE SURFACTANT. The process of

poisoned aggregation can also be characterized according to the structures and to the compo-sition of the final aggregates. This is interesting for a number of reasons.

.The aggregation of colloidal particles through Diffusion Limited Aggregation of Clusters

(DLCA) usually produces aggregates with fractal or porous structures [11-13]. Are such ten-

uous structures preserved in the final particles, or is there a reordering process by which the

voids collapse and the structures become dense?

.The amount of hydrophobic surface that is calculated from the particle radii is usually much

less than the amount that should have been protected by the adsorption of surfactant (see the

previous Sect.). Where is the wasted surfactant: has it been trapped inside the particles bythe reordering of subunits, or did it remain in the aqueous phase for kinetic reasons?

Quantitative answers to these questions were obtained through SANS, using the appropriatecontrasts to observe either the scattering from hydrophobic molecules and surfactants, or from

surfactants alone.

A first set of experiments was performed with particles made of cholesteryl acetate, with or

without PS-POE as a surfactant, in an aqueous phase made of D20. In this case the contrast

is between water and all the molecules that form the nanoparticles. These scattering curves

are shown in Figure 7: they show an initial downward curvature followed by a steep decay.The initial curvature is related to the particle sizes according to equation (3) the sizes measured


o~p

g

10°

fi

w~

_i Q~

. o1 ~

OZ

~

tl

io°cs/C~

Fig. 6. Surface area of the particles, divided by the surface area that would be kept if all surfactant

nlolecules were adsorbed on the particle surfaces. Hollow dianlonds: Cholesteryl acetate and PS-POE,

at cH =0.01 g lg. Filled squares: hexadecane and C12E5 at cH =

0.04 g lg. Hollow squares: hexadecane

and C12E5 at cH "0.1 g/g. The lines are a guide to the eye. Efficient use of the surfactant would

yield a surface area ratio equal to unity. Lower values indicate that sortie surfactant is wasted; highervalues indicate that other forces besides the adsorption of surfactant l1nlit the aggregation process.

through a fit of the scattering curves in this range of Q are in good agreement with those

obtained from QELS. The steep decay is related to the shape of the particles: according to

equation (4), the Q~~ Power law indicates that the particles are dense. Therefore, aggregationevents must be followed by a reordering process that collapses the voids to give dense structures.

A second set of experiments was performed with particles made of PSD, using PS-POE as

a surfactant, in an aqueous phase made of D20. In this case the contrast is between the

hydrogenated surfactant and the deuterated aqueous phase. The scattering curves obtained at

high ratios of surfactant to hydrophobic molecules (cs /cH"

0.5 orIi show a Q~~ decay, and

a characteristic shoulder near Q"

0.3 nm~~ (Fig. 8). The scattering curves of samples made

at low ratios (cs/cH"

0.05 to 0.25) show the same initial power law decay, but the shoulder

has vanished (Fig. 8).

A transcription of these features into real space is as follows. The Q~~ power law of the

initial decay comes from flat structures such as hollow shells. The shoulder at Q=

0.3 nm~l

shows that there are other objects that also contribute to the scattering; these objects have a

size on the order of I IQ A fit of the scattering curve is obtained by adding the scattering from

flat objects (Q~~ Power law) and the scattering from small spheres with a diameter of 7.6 nm.

These features are observed in dispersions where the coherent scattering is caused by the

surfactant molecules only. In terms of the location of the surfactant molecules, the interpre-

tation is as follows. The hollow shellsare

surfactant monolayers that cover the nanoparticles.


~io~ ~

*i

O ° O O

§p~3(

~~

d~ 10~ j

~slope=.4

~~i

io°

Q (nm~)

Fig. 7. Snlall Angle Neutron Scattering curves of dispersions containing cholesteryl acetate

nanoparticles in D20. Filled diamonds: pure cholesteryl acetate particles, without surfactant. Hollow

dianlonds: cholesteryl acetate particles with PS-POE as a surfactant, conlposition ratio cs/cH=

lcu=

io-~ g/g).

The change from Q~~ (Fig. 7) to Q~~ (Fig. 8) indicates that there is no surfactant in the

interior of the particles. The fit to the scattering in this range is obtained with shells that

have an average thickness of 24 nm and a standard deviation of the thickness of 5 nm.This

thickness is much larger than the thickness of a monolayer of PS-POE molecules, which is

calculated to be 8.2 nmfor the dispersion at cs/cH

"0.25 and 3.I nm

for the dispersion at

cs/cH"

0.10. These results show that the adsorbed surfactant does not form a monolayer of

uniform thickness, but rather a layer formed of thick and thin patches.

The excess scattering at Q"

0.3 nm~~ is observed in the dispersions that contain a large

excess of surfactant (Fig. 8); an obvious interpretation is that this scattering originates from

surfactant micelles that are in water. Indeed, the average molar mass of these micelles, mea-

sured through gel permeation chromatography, is 300000 g/mol. From this value, the aggre-gation number is calculated to be 150 PS-POE molecules, and the radius of the PS core is

calculated to be 3.9 nm. This is in good agreement with the size of the small spheres used in

the fit of the scattering curve(the water swollen POE shell would contribute only at smaller

Q values).

In sunlmary, the surfactant is located in irregular monolayers covering the particles and in

small micelles that are dispersed in water.


io~

io~

~~3

W~C

~10~

W~

fid~

~~o

I

i

~o

Q(nm ~)

Fig. 8. Scattering curves of dispersions containing PSD nanoparticles covered with PS-POE as a

surfactant, in D20. Hollow circles: Taking final concentrations equal to 5 x10~~ g/g for both PS-POE

and PSD (cs/cH=

ii; THF, 0.1 g/g. The full lines correspond to the calculated scattering curves for

a dispersion containing large shells (outer diameters beyond, thickness 24 nnl, standard deviation on

thickness, 5 nm) and snlall micelles (dianleter 7.6 nm). Filled circles: Concentrations equal to 5 x10~~

g/g and 10~~ g/g for PS-POE and PSD, respectively (cs/cH=

0.05) THF, 0.1 g/g. The dashed line

corresponds to the calculated scattering curve for a dispersion containing large shells (outer diameters

beyond 175, thickness 24 nnl, standard deviation on thickness, 5 nnl).

5. Numerical Simulations of the Growth Poisoning

5.I. THE MODEL. Numerical models for droplets growing by coalescence have been in-

troduced previously [24], however, the poisoning mechanism have not been accounted for in

these models. In the following, we introduce a lattice model to simulates a growth poisoningmechanism as exhibed by our experimental system.

5.I.I. The System. We have considered lattice sites limited in a cubic box of edge lengthL. Two particle species H (hydrophobic molecule) and S (surfactant) are considered. For the

starting configuration, time t=

0, a number NH of particles H and Ns of particles S, each one

of a cubic site size of unit length, are distributed randomly and uniformly in the box, avoidingoverlaps. Each particle is then a unit cell containing six faces of unit area. Concentration, or

volume fraction, of species are related to Ns and NH by:

cH =

~) (6a)L


cs ")~. (6b)

In our algorithm we introduce a time parameter r to include a time delay in which surfactant

(S specie) adsorption by cluster iH (formed by one ore more particles of the H species) becomes

possible.

5.1.2. The Algorithm. The algorithm consists in an iterative procedure, in which at a giveniteration a cluster iH or a surfactant is is picked up at random (under the assumption that

its diffusion coefficient is proportional to n°, a is the kinetic exponent equal to -) for a

3-dimensional compact system of n particles), according to probability distributions, for H

species:~j

~~" £~~ nj( + ns~~~~

and for S species

~~~ ~Jn~n

~ ~~~~~~

where ns is the number of free surfactants. Then, a displacement of a unit step is tried bychoosing at random a direction among the six possibilities (+1, 0, 0), (0, +1, 0), (0, 0, +1), taking

care of periodic boundary conditions (PBC). If the moving object does not try to occupy a site

occupied by other object, the displacement is performed. In the opposite case, no displacementis performed and: (I) if the collision is between a free surfactant and a surface site of specie H

(belonging to a cluster iH), a bond is established between them provided the time t is largerthan r, and the mass n~~ of the cluster iH is incremented by one. (ii) if the collision takes placesbetween two (or more) sites (H or S species) belonging to two (or more) different clusters, and

each one with at least one free surface site, we proceeds to the "coalescence" of the clusters byarranging the new mass (= n~>~ + n~ii~ +.. in a compact configuration with minimum surface.

Then, we arrange on the new surface the total number of surfactants isl + isii +.. adsorbed

previously by the coalescing clusters. If this number exceeds the new surface, the exceedingsurfactants are again distributed randomly in the empty sites of the box. (iii) no change is

made if the collision is between a free surfactant and another surfactant (even if its has been

adsorbed by the surface of a given cluster).After considering one of the above situations, the algorithm goes on by choosing randomly

another cluster or surfactant. The simulation is stopped at a given time tend in which all

clusters are saturated in surface by the joisoning effect, or if there is no more than one cluster

in the box. According to (7) the time t is calculated adding:

~~£~~

~~+ ns

~~~

at each iteration.

As the experiments described above, our numerical calculations have been addressed in the

diluted regime, i.e. when cH + cs < cp (cp is the percolation threshold). All results reportedbelow have been calculated at the end of the growth process (t

=tend)-

5.1.3. Two-Dimensional Illustration. We show in Figure 9 a two-dimensional representationof configurations obtained with our algorithm at three different stages of the process. For

t=

0, H (depicted in grey, or white when a cluster is saturated) and S (black sites) species

are distributed randomly in the box. For 0 < t < tend some clusters have been formed bycoalescence, and we observe that some of them

arefully saturated in surfaces by surfactants,


Fig. 9. Two-d1nlensional illustration of the growth poisoning for cH =0.06, cs = cH and L =100,

at three different stages of the process: for t=

0 (left), for 0 < t < tend (center), and for t=

tend

(right). Surfactant species (S)are depicted in black. The particles (or clusters) of hydrophobic species

(H) partially saturated by S species are depicted in grey, while they are depicted in white for fullysaturated particles (or clusters).

while other surface clusters are partially saturated. At this stage, even if there remains a

lot of free surfactants, the surfactant adsorption becomes hard because a given surfactant

must explore an important number of surface sites in order to find an available surface site.

For t=

tend the system reaches a stationary state since all clusters have saturated surfaces,impeding additional coalescence. Note that at this stage there remains some free surfactants.

In fact, we will show later that the concentration of the remaining free surfactants dependsstrongly on the ratio cs/cH.

5.2. RESULTS

5.2.I. The Average Cluster Volume Versus the Poisoning Ratio. In our three-dimensional

numerical simulation, we have calculated the average cluster volume using:

vav" j~j 19)

c

where nci is the number of saturated clusters. In Figure 10 we show a log-log plot of iv versus

the poisoning ratio cs/cH, for cH "0.001, r =

0 and three different box sizes L=

50 (circlesymbols), L

=80 (black diamonds) and L

=l10 (triangle symbols), and for

r =5000 and

L=

80 (open diamonds). This figure shows that the average volume V~v of the final clusters

decreases with increasing poisoning ratio cs/cH according to a succession of two power laws.

The first power law has an exponent -2.5, then there is a crossover at jf m 4 (r=

0) to a

slower decay, with an exponent -1.3. Interestingly, when there is a delay ~n the adsorption of

surfactants, I.e. r =5000, the crossover occurs at smaller values of cs/cH

,

but the exponentremains the same. In the following we discuss the absolute magnitudes of the cluster sizes, and

the way in which the surfactantsare used to control these sizes.

5.2.2. Finite Box Sjze and Lattice Effects. In the curves of Figure 10, the smallest cs/cHvalue corresponds to the case in which all clusters have coalesced in a single cluster of volume

iv=

NH- The corresponding plateau for smaller values of cs/cH is not shown in the figuresince it corresponds just to finite size effects, generally irrelevant for realistic applications. Also,

as deduced from the increasing box size L, this cs /cu value should tend to zero, when L -+ cc.


~~3

slope=-2.5

io~

io'

slope=-1.3

io°

cs/cH

Fig. 10. log-log plot of the quantity iv -1 versus the poisoning ratio cs/cH, for cH =0.001,

t=

tend, and for three different box sizes: L=

50 (circles), L=

80 (diamonds) and L=

110

(triangles). The corresponding time delay values are r =5000 for the hollow diamonds (L

=80),

while in the other curves we consideredr =

0. The full lines correspond to the different power law

domains that best fit our data. All these curves result from an average over 50 simulations.

5.2.3. Use ofsurfactants. The two different power laws observed in Figure 10, give infor-

mation about how surfactants are used. In fact, in a situation in which all Ns surfactants

are adsorbed by the cluster surfaces, the possible maximum number of saturated surfaces s*

should be equal to Ns, since, in general, each surfactant can saturate only one face particle of

unit area. Therefore, a normalized poisoned cluster surface § could be defined by:

§"

(10)S

where s is the number of unit area poisoned cluster surfaces. Note that, this quantity is

exactly equivalent to the per cent ratio of adsorbed surfactants and is indeed smaller than

unity. Furthermore, one has for distributions with well defined average size:

S* NS~

j.2/3 Ill)~nc] ~~

that combined with (9) gives:

~-3

iv c<~ (12)

CH

Note that the exponent measured in the first regime is close to -3, in contrast to the second

regime where it is approximately equal to 1.3. The slight deviation from the exponent -3

at low cs/cH may originate from surface roughness and from size polydispersity: both effects

increase the contents of the surfactant layers as compared with the contents of the particle

cores. Therefore, we can conclude that in the small cs/cH values the system is in a "efficient"


1.o

'

im

~~~

'

c~ /c~

Fig. 11. log-log plot of the normalized poisoned cluster surface § uerms the poisoning ratio cs/cH.The paranleters and legends are the saute that Figure 10.

regime of use of surfactants, in contrast to the observed "wasteful" regime at large cs /cH values,where the percentage of adsorbed surfactants decreases very quickly.

This is corroborated in Figure II where we show a plot of § vermscs/cH for same parameters

as in Figure lo. If all surfactants were adsorbed by the cluster surfaces, § should be equal to

one; from our results we note that in the regime of small cs/cH values § is close to one, and

decreases slowly when increasing cs/cH (this conclusion is still more convincing for L=

l10).In contrast, for large cs/CH values, § decreases very quickly when increasing cs/cH, and the

results for the different box sizes are practically superimposed.

The effect of a time delayr

between aggregation and adsorption is also shown in Figure II.

The corresponding curve (open diamonds) is below the curve at r =0, indicating that less

surfactant can be adsorbed on the particles and more is wasted. The crossover to the wasteful

regime occurs at lower cs/cH than that observed in Figure 10: this is because the normalized

surface § is strongly dependent on the size distribution function of the system (in contrast to

the quantity iv as defined from equation (9)) and, as we will show later, the shape of this

function becomes broader when increasingr.

5.2.4. Competition Between Poisoning and Coalescence. In this section we examine the

kinetic features of the competition between coalescence and poisoning. For this purpose, we

vary the initial concentration cH of hydrophobic molecules, and maintain a high poisoning ratio

cs/cH"

4. We show in Figure 12 how in all cases iv increases when cH increases forr =

0

(circles),r =

500 (squares) and r =2500 (triangles). In these simulations the variation of

cH land cs)was obtained by considering a fixed number of hydrophobic molecules NH

=750

(and surfactants Ns"

4NH), and varying the box size L as calculated from equation (6).With increasing r, the average cluster volume iv increases more and more quickly with cs,

these two quantities become proportional, as in a diffusion limited growth mechanism [11,12].


~~~

~

~~

~ ~

)D°

~

~

lJ

lJ

D~ ~ c

c~

Fig. 12. log-log plot of the average particle volunles iv versus the concentration cH, for a constant

poisoning ratio cs/cH=

4, and different time delay values:r =

0 (circles),r =

500 (squares) and

r =2500 (triangles). For comparisons the slope 1 is indicated by the full line. All these curves result

from an average over 30 simulations.

In fact, for larger

values the ~[v c< cH relation reflects mainly the initial stage of the growthmechanism where surfactant adsorption is forbidden.

To explain these results we have calculated the ratio sp/sc between the surfaces loss bypoisoning, sp, where:

i t~~~

sp =

/sp(t)dt (13a)

Nso

and the surfaces loss by coalescence:

sc =I sp. (13b)

Figure 13 shows sp/scas function of cH for the three different

rvalues and for a constant

poisoning ration cs/cH"

4. The decrease of sp/sc at r =2500 when cs increases reflects

the loss of surface by coalescence during the delayr.

The slight decrease of sp/sc at r =0

when cH increases reflects the fact that, in our model, coalescence is only diffusion limited,whereas surfactant adsorption is reaction limited. Indeed, at long times, a given surfactant

spends more time exploring the surface of a cluster to find an avalaible surface site than

performing a displacement to reach a cluster. Thus the increase of cH favors coalescence more

than adsorption, since the distances between clusters becomes shorter.

5.3. SizE DisTRiBuTioNs. We examine in this section the dependence on the poisoningratio cs/cH and the time delay

rof the size (or volume) distributions. In Figures 14 we show

the number nv of particles of volume V versw V for r =0 and two different poisoning ratios

cs/cH equal to 2 (black circles) and 4 (hollow circles) (see Fig. 14a), and for three different

time delay values (see Fig. 14b):r =

0 (circles),r =

2500 (squares) andr =

8000 (triangles).


1.o

O~ O o ~

D O O O OO

0.8

lJ

n

fl ~

o

uf ~ ~~

~~D

D~

~

0A~

~~

~~

~ ~

~'i000 0.002 0.004 0.006 0.008 °.°I °

c~

Fig. 13. Plot of the ratio of the surface lost by poisoning to the surface lost by coalescence sp/sc

versus cH. The paranleters and legends are the same as in Figure 12.

Note that in almost all cases the size distribution function nv Present two maxima, the first

corresponds to a monomer peak and the second illustrates the tendency of the system to form

particles with a "monodisperse" size distribution. The effect of concentration on the second

maximum (illustrated in Fig. 14a) is expected since at low poisoning ratio the average volume

should be bigger than at high poisoning ratio, as already observed in Figure 10. In Figure 14b

we observe that forr =

8000 the monomer peak vanishes, this result suggest that at high

rvalues the system main reflects the aggregation mechanism of a diffusion limited problem,

for which it is known that the cluster size distribution is bell shaped, presenting only one

maximum [11-13].The bimodal shape of the size distribution function is also manifested in our experimental

system as observed in Figure 3. The small peak at small size values indicates that the coales-

cence phenomenon is more important in the former bigger particles. These particles grow up

to reach a given average size which, as mentioned above, depends on the poisoning ratio. We

point out that the resulting bimodal shape observed here is also consistent with the resultingshape of the size distribution in a off-lattice model of droplets growing by coalescence [25].

6. Smoluchowski Approach of the Growth Poisoning

6.I. EQUATIONS. The main results of the preceding sections can be recovered using kinetic

Smoluchowski equations [26] of this growth process. We have first to assume that the fluctua-

tions of the densities (of surfactants and of hydrophobic molecules),can be neglected. Under

this assumption, the surrounding of each cluster or surfactant is the same and identical to the

average distribution of objects in the system. More precisely, we consider a collection of NH

molecules H, and Ns surfactants S. At a given time, a cluster of H molecules is defined byits size iH, its radius R~~ (which is as the power 1/3 of its size), the number E~~ of available

adsorption sites (which is as the power 2/3 of its size), and the number Sj~ of surfactants


1.5

jai

i.o

>c

0.5

~ ~0 20 40 60

V

2.0

(b)1.5

~

c I-o

o.5

o-O0 lo 20 30 40 50

V

Fig. 14. Size distribution curves for CH=

0.001 and L=

50. In la)r =

0, Cs=

2CH (black circles)

and Cs=

4CH (hollow circles). In (b) Cs=

4CH,r =

0 (circles),r =

2500 (squares) andr =

8000.

All these curves results front an average over 2000 s1nlulations.

adsorbed on its surface. At each time step, one investigates all the possible binary events:

H + H or H + S, leading to a different state by coalescence or adsorption. A probabilityweight is attached to each event, namely:

l~ ~ (14)Kin,jn~ ~ ~ ~ ~~~~ ~

~~


for coalescence of two clusters of radii R~~ and Rj~, provided at least one free adsorption site

is available on each cluster, and:

kj~ s "+ l) (R~~ + I) II

~~"(15)

' Rm EIH

for adsorption of a surfactant (assumed here to be of radius I). The terms I/R~~are due

to Stokes diffusion, and the R~~ + Rj~ are just the cross sections of the clusters during their

Brownian path. Then one event is chosen according to these probability weights, and the time

is increased by the amount:

dt= ~ ~~ ~

j16)

This leads to the kinetic equations:

dnj,s~ it)

dt~~~~~

~ I~J',J-J'"J',s~>11 j-j',s~_~i~ Kj',jllj',s~> llj,sj + kj,sllj,sj -illp kj,slip,sj lip

j',s~i,s~_~i j',s~i

=~?l~~ ~ ~JS"J,sj "p + ~ ~

~'~~ ~'~J J"SJ~'~ "~'~? "~ ~J

l~J + SJ Ej+j ). ji~~~

These equations seem too difficult to be handled analytically. An alternative way to studythem is Monte-Carlo simulations: the model is then quite similar to the model described in

the preceding section, except that there is no more geometry.

6.2. RESULTS. The advantage of the Smoluchowski approach is that it is simpler to imple-ment. There is no geometry, so the simulations are fast. Moreover, since one can write down

the kinetic equations, analytical investigation can be tempted in simple cases. Nevertheless,this approach is a mean-field theory of the model described in the preceding section, and its

validity for ordinary three-dimensional space is not ascertained. This validity has to be checked

on the results. In Figure 15, we have plotted the average cluster volume (Eq. (9)) versw the

poisoning ratio cs/cH, for T =0, and NH

"256. We see that the two regimes (small- cs/cH,

for which the surfactants are used efficiently, and large- cs/cH, that is excessof surfactants)

are well recovered. And we obtain the exponents -2.7 for the small- cs/cH values regime, and

-IA for the other one. Note that these two exponents are slightly larger than those obtained

in the Monte Carlo simulations, possibly as a consequence of the absence of geometry in the

Smoluchowski approach.The behaviour of all other computed data are quite similar and are not reproduced here. This

shows clearly that the Smoluchowski approach can be used in this process, and that the two

regimes meet at about the same point. For example, the ratio of the surface lost by poisoningto the surface lost by coalescence sp /sc, obtained from equation (13), decreases with cH (whqnaggregation becomes faster) and increases with cs /cH (when poisoning becomes more efficient).This last trend is illustrated in Figure 16: the rise of sp/sc with cs/cH is remarkably close to a

linear law. This can be explained in the following way.. At very low surfactant concentrations,most of the initial surface is lost by coalescence, therefore sc m aNH where a is the number

of surface sites in an H molecule (here,we choose a =

6.5 to match the values of number of

surface sites in the geometrical algorithm for cluster sizes up to 3). All surfactants are used to


~~3

slope=-2.7

~>$lope=-lA

~~o10

c~

Fig. 15. log-log plot of the quantity iv -1 versus the poisoning ratio cs/cH obtained from the

Snloluchowski nunlerical calculations. We considered NH=

256 and r =0. The full lines correspond

to the different power law donlains that best fit our data. This curve result front an average over 1000

simulations.

io'

slope=1 ~~o ~~~~Jf

~o

~~~~

~o

°

~ ,

io~

c s/c~

Fig. 16. Plot of the surface lost by poisoning to the surface lost by coalescence sp/scversus the

poisoning ratio cs/cH obtained from the Smoluchowski nunlerical calculations. The full line indicates

the slope1.


stop aggregation, therefore sp t Ns. Thus,

~~~

~~~f

~~ (~8)sc aNH cH

At very high surfactant concentration, the hydrophobic molecules never form clusters contain-

ing more than 2 molecules. In this limit the probability of forming an H-H bond is proportional

to cH, while that of forming an H-S bond is proportional to cs. Therefore the ratio sp/sc is

again proportional to cs /cH. The complete variation of sp /sc, shown in Figure 16, demonstrates

that aggregation is the dominant process in most practical situations (up to cs/cH"

4).

7. Discussion

We start this discussion with a brief summary of experimental results. These results are then

compared with the results from numerical simulations. Finally, we try to put together a model

for the role of the surfactant in precipitation processes.

7.I. SUMMARY oF EXPERIMENTAL RESULTS. The solution precipitation of hydrophobicmolecules in conditions where they are completely insoluble yields particles that are amor-

phous, dense and globular. Stable aqueous dispersions are obtained when the particles are

kept apart by adsorbed surfactants or by impurities that function as surfactants. In these

stable dispersions, the particle sizes are determined by two parameters: the poisoning ratio

cs/cH, I-e- the ratio of surfactant molecules to hydrophobic molecules, and the dilution ratio

cH, I.e. the ratio of hydrophobic molecules to solvent in the original solution.

The variations of particle sizes according to these parameters show that there are 2 regimes,which differ in the way surfactants are used. At very low concentrations of surfactants, the

surface area of the particles matches the area of a monolayer of the surfactant, indicating that

all surfactant molecules are used to keep particles apart and limit their aggregation. At higherconcentrations of surfactant, the surface area of the particles is much less than the area of a

monolayer of the surfactant (Fig. 6).Neutron scattering experiments on dispersions made in this regime reveal that part of the

surfactant is adsorbed on the particles, and part of it forms excess micelles in water.

Thus, attempts to reduce the particle sizes by adding increasing amounts of surfactant

become inefficient at some point where most of the added surfactant remains in water.

7.2. COMPARISON oF EXPERIMENTS wiTH NUMERICAL SIMULATION. The kinetic model

demonstrates the effects of the competition between aggregation of hydrophobic molecules and

poisoning by surfactant. The two-dimensional illustration presented in Figure 9 shows qual-itative agreement with the experimental results: particles covered with adsorbed surfactant

coexist with single molecules, also covered by surfactant, and with excess surfactant. Quan-titative results from the 3-dimensional simulation (Figs. 10 and II) show a regime of efficient

use of the surfactant followed by a regime where most surfactant is wasted. Comparison with

experiments (Figs. 5 and 6) shows that, in the simulation, the regime of efficient use extends

to much higher surfactant concentrations (cs/cH"

4 instead of 0.04); as a result, the par-ticle volumes have been reduced to extremely low values (taking I nm as the diameter of a

monomer, iv t 10 instead of 107 at the crossover). This discrepancy originates from the time

delay between aggregation and surfactant adsorption, as explained below.

In the efficient regime of this simulation, it was assumed that surfactant adsorption starts at

the same time as the aggregation of hydrophobic molecules. In the experiments, there is a time

delay between the two caused by the finite rate at which water diffuses into the solvent that

initially contains hydrophobic molecules and surfactants. A similar delay was introduced in


another simulation. At low surfactant concentrations, this simulation yields the same results

as the previous one because the initial aggregation leaves enough surface area to bind all

the surfactant (Fig. 10). However, at high surfactant concentration, the particle sizes remain

above those obtained with no time delay, because the initial aggregation is not prevented bythe excess surfactant. Comparison with the experiments shows that the actual time delay is

still much larger than that used in the simulations. A simulation in these conditions would

have to handle particles of extremely large sizes. Alternatively, the hydrophobic molecule used

in the simulation could be rescaled to be an aggregate containing 10~ molecules.

In the wasteful regime, the experiments show that the average volume per particle increases

linearly with concentration (Fig. 4). The numerical simulation reproduces this trend and pro-vides a simple explanation for it. Indeed, the particle volume rises linearly with concentration

only in the limit of long delays (Fig. 12). Therefore the rise in the final volume reflects the

unimpeded aggregation during the time delay before the beginning of surfactant adsorption.The linear rise with concentration is indeed that expected from the Smoluchowski equation for

kinetic aggregation [I Ii.In conclusion, the results of experiments are well-reproduced by simulation if aggregation

is allowed to proceed unimpeded for a time t, and then adsorption of the surfactant starts.

At low surfactant concentration, there is, at that point, more than enough surface to adsorb

all surfactant molecules; thus aggregation continues until it is limited because all remaining8urfaces are poisoned. Thus the surfactant molecules are used efficiently, but the final parti-

cle volumes are large. At high surfactant concentrations, adsorption of surfactant stops the

aggregation immediately after the time delay for adsorption. Consequently, the aggregationnumbers are determined by the length of this time delay and by the rate of aggregation, I.e.

by the concentration of hydrophobic molecules. Adding more surfactant does not change these

particle volumes; consequently the excess surfactant is wasted.

7.3. MODEL FOR THE PRECIPITATION PRocEss. As in any kinetic process, the precipita-tion of hydrophobic molecules in water can be divided into three stages: initiation, propagationand termination.

Initiation is caused by the mixing of water with the solution containing the hydrophobicmolecules and surfactant. Immediately, the hydrophobic molecules aggregate, because they

are insoluble in this mixed solvent. At this stage, however, the surfactants do not bind to the

growing aggregates, because they are not sufficiently hydrophobic.

Propagation proceeds through random collisions of the growing aggregates. According to

the Smoluchowski equation, the average volume grows linearly with time, at a rate that is

proportional to the concentration of hydrophobic molecules.

Termination occurs when the water content is so high that the surfactants adsorb on the par-

ticles. The time delay between aggregation and termination sets the final size of the particles.

In practical situations, it is usually desired to obtain dispersions that are as fine as possible.The results described above set severe limitations for this endeavour. Increasing the surfactant

concentration beyond the amount needed to cover the particles at the end of the aggregation

stage will not be efficient, since the excess surfactant will just be wasted. Diluting the initial

solution will reduce the rate of aggregation, and therefore the particle sizes; however, the

final dispersion may be so dilute that it will be necessary to reconcentrate it, which may

cause further aggregation. Reducing the time delay between aggregation and adsorption of

surfactant requires a device that achieves instantaneous mixing of water and polar solvent. If

a high pressure homogenizer could be adapted to achieve this instantaneous mixing, it would

be the most attractive option.


Acknowledgments

This work used the neutron beams of ILL in Grenoble, France. It was performed as part of

the BIOAVENIR program financed by RHONE-POULENC with the MRE and MICE. One of

us (A.H.) would like to acknowledge support from CONICIT (Venezuela) and CNRS (France).

References

[Ii Puisieux F., Barratt G., Couarraze P., Couvreur P., Devissaguet J-P-, Dubernet C., Fattal

E., Fessi H., Vauthier C. and Benita S., Polym. Biomater. (1994) 749.

[2] Al-Angary A. and Halbert G. W., Drug Targeting Delivery (Microencapsulation of drugs)1 (1992) 277.

[3] Allemann E., Gurny R. and Doelker E., Eur. J. Pharm. Biopharm. 39 (1993) 173.

[4] Gregoriadis G., Liposomes as drug carriers (Wiley, 1988).

[5] Julienne M.C., Alonso M.J., Gomez Amoza J.L. and Benoit J.P., Drug Dev. Ind. Pharmacy

18 (1992) 1063.

[6] Koosha F. and Muller R.H., Archiv. Pharm., suppl. 321 (1988) 680.

[7] Sj6str6m B., Kaplun A., Talmon Y. and Cabane B., Pharm. Res. 12 (1995) 39.

[8] Spenlehauer G., Vert M., Benoit J.P., Chabot F. and Veillard M., J. Controlled Rel. 7

(1988) 217.

[9] Alkhouri-Fallouh N., Roblot-Treupel L., Fessi H., Devissaguet J-P- and Puisieux F., Int.

J. Pharmace~t. 28 (1986) 125.

[10] StoInik S., Davies M.C., Illum L., Davis S.S., Boustta M. and Vert M., J. Controlled Ret.

30 (1994) 57.

[I Ii Jullien R. and Botet R., Aggregation and Fractal Aggregates (World Scientific, Singapore,1987).

[12] Meakin P., Phase Transitions and Critical Phenomena, vol. 12 (Academic Press, New

York, 1988) p. 335.

[13] Vicsek T., Fractal Growth Phenomena (World Scientific, Singapore, 1989).

[14] Wendorf J.H. and Price F.P., J. Phys. Chem. 75 (1971)18.

[15] Wendorf J.H. and Price F.P., Mol. Cryst. Liq. Cryst. 25 (1974) 71.

[16] Kunihisa K.S. and Gotoh M., Mol. Cryst. Liq. Cryst. 42 (1977) 97.

[17] Cotton J.P., Neutrons, X-rays and Ligth Scattering, P. Linder and Th. Zemb, Eds. (Else-vier Science Publishers, 1991) p. 3.

[18] Strazielle C. and Weill G., Analyses chimiques et caractAiisations, Techniques de

l'IngAnieur PI, pp. 1065-1-1065-22.

[19] Cabane B., Surfactants solutions: new methods of investigations, Surfactants Sciences

Series, R. Zana, Ed., vol. 22 (Marcel Dekker Inc., New York, 1987) p. 57.

[20] Bellare 3.R., Davis H. T., Scriven L. E. and Talmon Y., J. Electron Microz. Techniq~e10

(1988) 87.

[21] Lichterfeld F., Schmeling T. and Strey R., J. Phys. Chem. 90 (1986) 5762.

[22] Olsson U., Wlrz U. and Strey R., J. Phys. Chem 97 (1993) 4535.

[23] Sonneville O., private communication.

[24] for a review see: Meakin P., Rep. Prog. Phys. 55 (1992) 157.

[25] Family F. and Meakin P., Phys. Rev. A 40 (1989) 3836.

[26] van Dongen P.G.J. and Ernst M.H., J. Stat. Phys. 50 (1988) 295.

Documents

Surfactant Limited Aggregation of Hydrophobic Molecules in